Problem: Determine how many solutions exist for the system of equations. ${-4x-2y = 4}$ ${-18x-3y = 9}$
Solution: Convert both equations to slope-intercept form: ${-4x-2y = 4}$ $-4x{+4x} - 2y = 4{+4x}$ $-2y = 4+4x$ $y = -2-2x$ ${y = -2x-2}$ ${-18x-3y = 9}$ $-18x{+18x} - 3y = 9{+18x}$ $-3y = 9+18x$ $y = -3-6x$ ${y = -6x-3}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = -2x-2}$ ${y = -6x-3}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.